1-4244-1461-X/08/$25.00 ©2008 IEEE

A Study on Optimal Age Replacement of Dozer Engines

P.V.N.Prasad, Ph.D., Osmania University K.R.M.Rao, Dr. Inz., Osmania University

Key Words: Total time on test , Renewal Process , Non-Homogeneous Poisson Process (NHPP), Kaplan-Meir Estimate , Nelson Estimate, Maximum Likelihood Estimate

SUMMARY & CONCLUSIONS

In this paper, the concepts and methods of total time on test (TTT) plots are discussed. The failure data of two groups of electro-mechanical equipment (indigenous and imported dozer engines) are used to draw TTT plots, which indicate health of equipment in terms of failure rate. The optimum replacement interval is obtained for the equipment with increasing failure rate depending on the type of the data i.e. independently and identically distributed (i.i.d) or not. Graphical and analytical tests are used to confirm the nature and trend of the data. Non-parametric methods like Kaplan-Meir estimate and Nelson estimate are used to evaluate cumulative distribution function and scaled TTT to construct the plots. Parametric methods like Weibull and Power Law Process models are used to verify the graphical results.

1 INTRODUCTION

The concept of total time on test (TTT) plots introduced by Barlow was proved to be a very useful tool in many reliability applications such as model identification, aging properties, age replacement policies etc. Plotting data is the first step to understand the information contained in the data. In case of failure data of repairable equipment, these plots indicate increasing or decreasing or constant failure rates. The health monitoring of any equipment can be found using these plots, whether the failure data satisfies i.i.d assumption or not. If the data confirms i.i.d assumption, optimal age replacement interval is obtained by plotting the scaled TTT with cumulative distribution function (CDF). The CDF provides the cumulative probability of failure. If the data shows any trend in the form of deterioration, optimum replacement interval is obtained from the TTT plot by considering age of the equipment as discussed by Barlow & Campo [1]. For deciding the optimal replacement / overhaul interval based on age replacement, the ratio of cost of planned maintenance to the additional cost due to failure of equipment over planned maintenance cost is the criteria, if the data represents renewal process. However, if the data is represented by Non-Homogeneous Poisson Process (NHPP) model, the ratio of planned maintenance cost to the failure maintenance cost is the deciding factor. When the TTT plot is above the diagonal line, increase in failure rate is indicated. Analytically it can be confirmed by the value of shape parameter of Weibull / PLP model, which is more than unity.

Uday Kumar et al analyzed several Loading machines using TTT plots to identify the tendency of their failure rates [2]. Ulf Westberg and Bengt Klefjö [3] used theses plots for model identification of failure distribution models and also to determine the replacement time for a repairable system on some simulated data and censored failure data of sheet metal leveler [4]. Dhananjay Kumar and Ulf Westberg [5] obtained replacement of ages of pressure gauges for different levels of pressures. David Reinke, Edward Paul and Paul Murdrock [6] used a highly censored simulated data of series system to estimate optimal replacement time. Barlow [7] presented the usage of TTT plots for non-stationary data with NHPP model.

This paper presents the method of estimation of optimal age replacement of field data at certain cost ratios. The optimal replacement time is the time, which minimizes the average maintenance cost per unit time in the long run. The failure data of a two groups of electro-mechanical equipment i.e. an indigenous dozer engine and an imported dozer engine, collected over a period of eight years is chosen for the present study. Different non-parametric methods like KME method, Nelson method are adapted to construct TTT plots. The results are compared with the results obtained by parametric TTT transforms using maximum likelihood estimates (MLEs) of Weibull distribution or Power Law Process (PLP), depending on the absence or presence of trend in the data.

2 TTT TRANSFORMS

Let 0 ≤ t1 ≤ t2 ≤………..≤ tn be an ordered sample of size n from a cumulative life distribution F(t) with F(0) = 0, survival function R(t) = 1- F(t) and finite mean given by [8]

)(0∫∞

= dttRμ (1)

The TTT transform is defined as,

∫=it

i dttRt0

)()(μ (2)

with mean value μ and μ(0) = 0. The scaled TTT transform

is defined as, μϕ ∫=it

i dttRt0

)()( (3)

The curve between ϕ(t) vs. F(t) is the TTT transform and it is the graphical representation of the analytical solution for parametric models.

3 OPTIMUM REPLACEMENT POLICY

An important application of TTT plot is determination of optimum replacement schedule of equipment under study. The failure data of equipment can be renewal model type or non-stationary model type and the following sections give the methods of constructing TTT plots for these types of data.

3.1 Renewal Data

In this policy, the equipment is renewed or replaced at a cost Cf at failure or at a cost Cm at a planned replacement. The later renewal / replacement occurs when the equipment has reached a certain age. The purpose is to find the optimum replacement interval To at which the long-run maintenance cost per unit time is minimum and is given by [3,4]

0 0

( ) [1 ( )] ( )( )

[1 ( )] [1 ( )]f m f

T T

C F T C F T C aF TC T

F x dx F x dx

+ − += =

− −∫ ∫ (4)

The equation can be modified by taking Cf – Cm = a, where 'a' is the additional cost incurred due to failure over maintenance cost and defining a constant K = a / μ )(/)]()/[()( ttFaCmKTC φ+= (5)

The plot of ϕ(t) vs. F(t) is the TTT plot. The cost is minimum, when the slope of the plot i.e., φ(t)/[(Cm/a)+F(t)] is maximum. If a tangent with largest slope is drawn from (-Cm /a, 0) on the plot, the optimum replacement interval can be obtained. But, if it touches the plot at (1,1), there is no optimal solution.

3.2 NHPP Data

The TTT plots can also be used in special cases of dependent observations [7]. If the time between failures are not independent, but depend on age of the equipment then the number of failures N(t) in [0,t] has Poisson process. If we have failure histories for a group of similar equipment, then we can pool the observed ages. Let t1 ≤ t2

≤…………….tN(T) be the ordered ages and n(t) be the number of processes under observations at system age t. Then, the scaled TTT for NHPP model is a plot of

∫∫)tn(N

0

i

0

tdt)t(n

tdt)t(n vs. CDF.

Let C1 be the average cost of repairing on failure and C2 be the cost of new replacement. If a replacement is made at time t, the long run average cost of replacement/ overhaul is,

C(t) = [C1.E(t) + C2] / t (6) where E(t) is the expected no. of failures. The numerator is the expected cost of life cycle of length 't'. To determine, optimum replacement age T* graphically, the value -C2/C1E(T) is located on x-axis of TTT plot. A tangent is drawn with maximum slope from that point to the plot and projected it on x-axis. The corresponding time is the optimum replacement time T*.

4 TTT PLOTS Different methods can be used to estimate the survival

function for the construction of TTT plots. Kaplan-Meir Estimate (KME) and Nelson methods are important non-

parametric methods for the estimation of survival function of an uncensored data.

4.1 KME Method

For uncensored data, the survival or reliability at a failure time ti is given by [9], R (ti) = (n - i) /n (7)

where, i = 1,2,………….n and 'n' is the total no. of failures. The TTT is given by,

111

)( −−=

−= ∑ jji

jj RttTTTi (8)

where T0 = 0, R(0) = 1 and R(tn) = 0

4.2 Nelson Method

In this method, the function is continuous between two successive failures and assumed to follow exponential distribution [10]. The hazard rate is given by,

hi = 1/[(n + 1 – i)(ti – ti-1)] (9) and the cumulative hazard function is,

∑=

−+=i

ji jnH

1)1/(1 (10)

The survival function and TTT are given in the following equations.

Ri = exp (-Hi) (11)

iji

jji hRRTTT /)(

11 −= ∑

=− (12)

5 ANALYTICAL MODELS

5.1 Weibull Model

In a two-parameter Weibull distribution, the reliability is given by,

( ) exp[ ( ) ]tR t δ

α= − (13)

The increasing failure rate (δ >1) necessitates evaluating optimal preventive maintenance interval at different cost ratios. The optimal replacement value of T0 is obtained from setting its first derivative equal to zero. Thus, rearranging Eq.(4) and performing differentiation with respect to T, to find the optimum maintenance interval, we get the following equation.

0( ) ( ) ( ) 1

TmCR T T R x dx aλ+ − =∫ (14)

where λ(T) is the failure rate or hazard function of two-parameter Weibull distribution, which is given as,

( ) ( )1

( ) tt δδλ α α= (15)

For increasing failure rate, we must have δ >1 and hence the Eq.14 becomes

( ) ( ) ( ) ( )1

011

[ 1]

xT T

f

m

Te e dx CC

δ δδα αδ

α α−− −

+ − =−

∫ (16)

The above equation gives optimum value of preventive maintenance interval. The above non-linear equation can be

solved, by numerical integration by assuming,δ

α∫⎟⎠⎞

⎜⎝⎛−

=T

T

ex0

5.2 Power Law Process Model

Most of the repairable equipment, especially those working for a long time, show the presence of strong trend. Such systems can be analyzed by using the NHPP model, which assumes that time between failures, varies as a function of time. One type of NHPP model, which can be used to model the trend, is the Power Law Process (PLP) model, where failure intensity, v(t) is given by [11,12]. v(t) = (δ / α). (t / α) δ -1 (17) where, α and δ are scale and shape parameters respectively. The parameters can also be estimated by using the following analytical expressions.

1 1

1

,ln

nn

n

ii

t ntn

tδ

α δ −

=

= =⎛ ⎞⎜ ⎟⎝ ⎠∑

(18)

where, n is the number of failure events, ti is the total running time at the occurrence of ith failure and i = 1,2,3…n. In this estimation, α is the time for first failure, which may not be of much importance for repairable items. But δ provides information about failure pattern of repairable system. For repairable equipment, with minimal repair the failure history can be modeled by the PLP model with δ >1 [13]. From the intensity function, the cumulative number of failures is given by mean value function [7,11]. E[N(t)] = (t / α )δ (19)

If N(t) is the cumulative number of failures at time t, then N(t) can be equated with E[N(t)]. The optimum maintenance interval is determined using Eq.6 and Eq.19, and the following expression can be obtained.

δ

δα

/112

1/

* ⎟⎠⎞

⎜⎝⎛

−=

CCT (20)

6 ANALYSIS OF FAILURE DATA FOR REPAIRABLE SYSTEM

Three basic models normally used in repairable systems are HPP, RP and NHPP models [14]. Firstly, the data should be checked for evidence of trend. If no trend is detected, then one should proceed by analyzing the inter-arrival times. A goodness-of-fit test can be used to determine if an exponential distribution is appropriate, otherwise a more general renewal process must be fitted like Weibull distribution. If evidence of trend is concluded, a non-stationary model must be fitted. The Power law Process is one the most commonly used non-stationary models, within NHPP models.

6.1 Trend Test

Graphical and analytical tests are used to detect trend in the data. The most common visual test is a plot of cumulative number of failures vs. cumulative time. If the graph wriggles around the diagonal line, then absence of trend can be

assumed. Otherwise, if the graph is convex (concave), presence of trend or improvement (deterioration) can be assumed. Analytical tests like Laplace test and Modified Laplace test can be performed to check for HPP and RP assumptions respectively. As the maintenance interval is to be planned for equipment with increasing failure rate (IFR), there is need to check if the data is renewal process.

6.2 Laplace Test

The null hypothesis of this test is HPP. If T1, T2,………….,Tn are chronologically ordered arrival times, then the test statistic UL is given by [15],

∑∑∑∑===

−

=−−−=

r

qqnqqn

r

qq

r

q

n

iqiL TnTnTU

1

2121

121

1

1

1)1(/)1( (21)

where, r is the no. of units failed, Tqi is the time of the ith failure for the qth item and nq is the no. of failures for the qih

item. At a s-significance level α there is evidence of trend if UL > zα /2 for deterioration and UL < -zα /2 for improvement, where zα /2 is the upper critical value in a standard Normal distribution.

6.3 Modified Laplace Test

When the null hypothesis is renewal process, if μ is the mean and CV is the co-variance of the data, then the test statistic ULR is given by [15],

ULR = UL / CV (22) where, the square of co-variance is given by,

22

11

12 /)( μμ−= ∑=

−

n

iin TCV (23)

The s-significance levels discussed in section 6.2 are valid to test and confirm for RP model. The statistic ULR also becomes approximately standard Normal, under the null hypothesis. This test is also called Lewis-Robinson test.

7 ANALYSIS OF DOZER ENGINES

The failure data of two groups of dozer engine along with TTT calculations are shown in Table 1 & Table 2. Visual examination of the data revealed that indigenous engine, with a very weak trend could be fitted in a renewal process model and imported engine with a substantial trend of deterioration can be fitted in a NHPP model. These are shown in Figure 1 and Figure 2 respectively.

The statistical tests are performed on the engine data at a significance level of 5% to confirm the assumption obtained from trend tests. The details and results are shown in Table 3. The failure data of both types of engines on dozers is shown in the Appendix. TTT plots drawn for indigenous engines, using KME and Nelson methods are shown in Figure 3 and Figure 4 respectively. The ML estimates are found for Weibull distribution and age replacement interval is evaluated analytically. For imported engines, the plots using the two non-parametric methods are shown in Figure 5 and Figure 6. The Power Law Process is assumed and the ML estimates are evaluated. To confirm the assumption of Weibull model for indigenous engine and PLP models for imported engine, Mann's test and Cramer-von-Mises test are performed on the

respective engine data at significance level of 5%. The details are shown in Table 4. The analytical block replacement interval for imported engine can be found at an assumed cost ratio and the results along with ML estimates are shown in Table 5.

KME Nelson

Time (Hrs.)

CDF Scaled TTT

CDF Scaled TTT

000 0.00 0.000 0.000 0.000 055 0.04 0.055 0.039 0.053 158 0.08 0.153 0.078 0.150 342 0.12 0.321 0.118 0.314 346 0.16 0.324 0.157 0.318 464 0.20 0.423 0.196 0.414 527 0.24 0.473 0.235 0.463 533 0.28 0.477 0.274 0.467 663 0.32 0.570 0.314 0.558 694 0.36 0.591 0.353 0.579 698 0.40 0.593 0.392 0.581 799 0.44 0.654 0.431 0.640 883 0.48 0.700 0.471 0.686

0991 0.52 0.756 0.510 0.741 1056 0.56 0.787 0.549 0.771 1103 0.60 0.808 0.588 0.791 1105 0.64 0.808 0.627 0.792 1130 0.68 0.817 0.666 0.801 1152 0.72 0.824 0.706 0.807 1212 0.76 0.841 0.745 0.824 1244 0.80 0.849 0.784 0.831 1402 0.84 0.880 0.823 0.862 1718 0.88 0.930 0.862 0.911 1874 0.92 0.949 0.901 0.929 2012 0.96 0.960 0.940 0.940 3029 1.00 1.000 1.000 1.000

Table 1 - TTT Calculations of Indigenous Engine

KME Method Nelson Method

Time (Hrs.)

CDF Scaled TTT

CDF Scaled TTT

0000 0.0000 0.0000 0.0000 0.0000 2410 0.0588 0.2542 0.0540 0.2431 3691 0.1176 0.3811 0.1081 0.3651 5169 0.1765 0.5181 0.1621 0.4976 6140 0.2353 0.6019 0.2162 0.5792 6393 0.2941 0.6222 0.2702 0.5991 8005 0.3529 0.7407 0.3242 0.7165 8320 0.4118 0.7618 0.3783 0.7377 8790 0.4706 0.7904 0.4323 0.7667 9575 0.5294 0.8330 0.4863 0.8106 9980 0.5882 0.8524 0.5403 0.8311

11125 0.64719 0.9000 0.5943 0.8824 13793 0.70592 0.9938 0.6483 0.9869 13824 0.7647 0.9947 0.7023 0.9888 13984 0.8235 0.9983 0.7563 0.9925 14059 0.8823 0.9995 0.8102 0.9941 14082 0.9412 0.9997 0.8640 0.9945 14174 1.0000 1.0000 1.0000 1.0000

Table 2 - TTT Calculations of Imported Engine

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000 Cum. Time (Hrs.)

Figure 1 - Trend Plot of Indigenous. Engine

Cum

. Fai

lure

s

0

4

8

12

16

0 2000 4000 6000 8000 10000 12000 14000Cum. Time (Hrs.)

Figure 2 - Trend Plot of Imported. Engine

Cum

. Fai

lure

s

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1Figure 3 - TTT Plot of Indigenous Engine - KME

Scal

ed T

TT

CDF

Table 3 - Statistical Details of Dozer Engines

Table 4 - Statistical Tests

Engine Type Indigenous Imported

Cost Ratio Cm /a = 0.5 C2 /C1=1.0 KME 1056 5169 Nelson 1056 5169

Opt. Rep. Interval (Hrs.) MLE 1113 (Weibull) 6737 (PLP) Parameters α, δ 1121.7, 1.60 7657.8, 2.35

Table 5 - Optimum Replacement Interval

APPENDIX

The failure data of imported engines (D1 to D7) and indigenous engines (D8 to D22) are given in Table 6.

S.No. Eqpt. No. Life in Working Hours 1. D1 2410, 3983, 2397, 5003 2. D2 8320, 5504, 160, 190 3. D3 6140, 1865, 3120, 2934 4. D4 3691, 5884, 405, 4102 5. D5 Still Working 6. D6 5169 7. D7 Still Working 8. D8 1402, 1212 9. D9 55, 663, 158, 694 10. D10 3029 11. D11 1874 12. D12 Still Working 13. D13 1718 14. D14 883, 1056 15. D15 464, 991, 698, 342 16. D16 1152 17. D17 346, 1103, 1105 18. D18 1244 19. D19 2012 20. D20 799 21. D21 527, 1130 22. D22 533

Table 6 - Failure Data of Dozer Engines

ACKNOWLEDGMENTS

The authors would like to thank the Head of Electrical Engineering Department, University College of Engineering, Osmania University for the support and co-operation. Special thanks are due to Dr.David M.Reineke of Air Force Institute of Technology, Dayton for his timely suggestions

Details Engine Type Tests Indigenous Imported Null Hypothesis Weibull PLP Test Type Mann's Test CVM's Test Test Statistic 0.816 0.186 From Tables F0.05, 24, 24 =1.984 M0.05, 16 = 0.216Remarks Accepted Accepted

0 0.2 0.4 0.6 0.8

1

-0.5 0 0.5 1Figure 4 - TTT Plot of Indigenous Engine - Nelson

Scal

ed T

TT

CDF

0

0.2

0.4

0.6

0.8

1

-0.2 0.3 0.8CDF

Scal

ed T

TT

Figure 5 - TTT Plot of Imported Engine - KME

0

0.2

0.4

0.6

0.8

1

-0.2 0.3 0.8CDF

Scal

ed T

TT

Figure 6 - TTT Plot of Imported Engine - Nelson

REFERENCES

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The Logistics Institute, Univ. of Arkansas, 1996 BIOGRAPHIES

P.V.N.Prasad, B.Tech., M.E, Ph.D. Assoc. Professor, Dept. of Electrical Engineering. Osmania University Hydearbad – 500 007, India. E-mail: polaki@rediffmail.com

P.V.N.Prasad graduated in Electrical & Electronics Engineering from Jawaharlal Nehru Technological University, Hyderabad in 1983 and received M.E in Industrial Drives & Control from Osmania University, Hyderabad in 1986. He obtained his Ph.D in 2002 from Osmania University. He served as faculty member in Kothagudem School of Mines (1987-95) and at presently serving as Assoc. Professor in the Department of Electrical Engineering, Osmania University, Hyderabad. His areas of interest are Reliability Engineering, Proportional Hazard Modeling and Computer Simulation of Electrical Machines & Power Electronic Drives. He is a member of Institution of Engineers (India) and Indian Society for Technical Education. He is recipient of Dr.Rajendra Prasad Memorial Prize, IE (I), 1993-94 for best paper. He has got over 35 publications in National Journals & Magazines and International Conferences & Symposia and presented technical papers in India, Thailand and Italy. K.R.M.Rao, B.E., M.Sc., (Engg.), Dr. Inz Professor, Dept. of Electrical Engineering. M.J.College of Engg. & Technology Hyderabad – 500 001 E-mail: drkrmr@yahoo.com

K.R.M.Rao graduated in Electrical Engineering from Andhra University in 1962, received M.Sc. (Engg.) from BHU in 1964 and Doctor of Technical Sciences degree from Wroclaw Technical University, Poland, 1968. He served as Assistant Professor in Electrical Engineering during 1969–74 at H.B Technological Institute, Kanpur. Later, he served as Professor in Department of Engineering and Mining Machinery, Indian School of Mines, Dhanbad (1974–78), Professor, Kothagudem School of Mines (1978 - 95), Professor of Electrical Engg., Univ. College of Engg., and Dean, Faculty of Engineering, Osmania University, Hyderabad (1997–99). Presently Dr.Rao is with Department of Electrical Engineering, M.J.College of Engg. & Technology, O.U, Hyderabad. His areas of interest are Mining Machinery, Material Handling, Mining Electrical Engineering, Reliability Engineering and Maintenance Management. He is a Fellow, Institution of Engineers (India), Member, ISTE, Member, MGMI and Member, SRE (India).